Archive for the 'Calculus' Category

A message last night from a friend pointed out that this chapter may contain too much and might be unfocused. It was a great suggestion. I had just sent a note off to my editor saying that I had decided that chapter two would be about limits and that continuity would now become chapter three.

The down side is that this seems to extend the discussion of these two topics while the student is raring to go and wants to get to the good stuff. As they will later learn, however, this is the good stuff. So many of the theorems require an understanding of these two topics. The one I was tempted to give short shrift to is continuity – we’ll see.

As for limits, I’m leaving the formal definition until the end of the chapter. This means that I’m not providing formal proofs of any of the limit theorems. That may be ok. I want students to understand that if the limit as x -> a f(x) = 7 then lim x->a [f(x) + 3] is 10. I’ve seen tons of students who know the limit of the sum is the sum of the limits when they are studying that section of the book – can we get them to grasp these theorems at a gut level.

It also sets up nicely the expectation that the derivative of the product is the product of the derivatives.

I apologize for the following personal note in the midst of a Calculus blog.

Writing a book is a large commitment. You have to have a bigger reason than possible financial reward for writing. And I did. I looked at my beautiful, bright daughters and wanted them to have this book when they were ready to explore Calculus. Chris Adamson, friend and co-editor on several O’Reilly sites said something similar when I was wavering on writing it. He wanted to put it away for his children.

The number of things we have put away for our children.

Maggie Rose, my eldest, has already proofread the first chapter and given it her approval. She’s nine but I truly value her opinion. Elena Maxine loved the pictures. She loved that her dad could write books. There was so much that she loved and expressed. She was a beautiful, happy, loving child who died yesterday afternoon suddenly.

As Kim and I sat up last night crying and holding each other, I asked what the point was of continuing the book. She said “you need to dedicate it to her”.

I sobbed.

I’ve co-written nine books now and never dedicated one to my children. This was to have been the one. I suppose it will be.

I will need help writing this. I will need help for lots of things I suppose. Yesterday it didn’t feel real. Today it feels insurmountable.

It’s certainly a discontinuity. It’s odd because it oscilates between feeling like a vertical asymptote that I’ll never surmount and a removable discontinuity – she’ll wake up soon and be with us again. Of course it’s a jump discontinuity. Something dramatic happened yesterday and life is changed. I’m just not sure how yet.

I don’t know if this is common to mathematicians – but having a deep understanding of the infinite makes me profoundly afraid of death.

In any case, I hope you will forgive this personal intrusion. I have started a personal blog to work this through at Dear Elena.

There are people who have taken Calculus one or more times and still don’t have a feel for what it is they know. They can solve specific problems in context. They have learned which word problems are like which of the completely solved examples in their text. They know how to curve sketch because the rule says that if in this region the curve has a positive first derivative and a negative second derivative then the curve must look like this.

I took Calculus twice. Once while I was in high school and once in college. I could have skipped it the second time but I had an advisor smart enough to recommend otherwise. The first time I learned to do the problems and the second time I had time to find out a bit about what was really going on.

When I started working on HF Calc, I contacted friends who had been teaching Calculus for a while. One of them, Chris Butler, has reminded me that there are people in another camp as well. There are people who understand the theory and principles but can’t solve problems to save their lives. For them the algebra has no meaning. The symbols on the page don’t seem to be related at all to the graphs their calculators produce.

I’m sure the seeds are sown early. Some teachers work better with one group or another.

My nine year old daughter is just beginning to solve algebra problems. When she gets something like

x + 12 = 30

she writes

x = 18

. The teacher writes her notes that she will receive no credit for this work without showing the steps. She is to write:

x + 12 = 30
x + 12 - 12 = 30 - 12
x = 30

For Maggie, this is fine. It doesn’t hurt her to show her work and would have helped the teacher identify what she’s doing wrong in cases that she doesn’t get the right answer.

But there are plenty of people who memorize the steps and don’t feel the problem.

The title of this blog is “Extreme Teaching”. It comes from parallels I’ve long felt with Extreme Programming which I talk about in other posts. But you will hear people talk about code and how the programs they are working on push them in a particular direction. The code suggests a simplification. You’ll hear novelists talk about the characters they create in the same way. This character that springs entirely from their own head now seems to have a life of her own.

The same is true in even the simplest of algebra problems. There is a tension in the equation

x + 12 = 30

This tension wants to be resolved. The resolution is, what we would call in XP, a refactoring. In refactoring we initially go slowly until we have confidence in what we are doing. So initially we would substract 12 from both sides by explicitly writing it out. It’s how we can see that we truly didn’t change the equality. Later we just “bring the 12 to the other side”.

I’m not sure if people who learn this way are in the first or second group. I think it’s a mix.

I worry about the ones who are convinced that they can not do mathematics. The ones who hate mathematics are a different story for a different day. But those who have a hard time with calculations,are really bothered by
little mistakes the teacher makes on the board.

When I used to teach, I used to keep colored chalk on hand to highlight and annotate the body of notes. If I was working through a calculation and something interesting happened to get us from one step to the next – it might require a dream sequence. In the classroom, the students hear what the teacher is saying and understand what they are doing – but later their notes are missing all of the annotations (unless the teacher specifically wrote them down) and may not be able to reconstruct what was done.

I often taught concepts on more than one level to reach people from both camps. Different people learn differently. I’m certain that Head First Calc will speak to the people in the first camp and I’m working hard to make sure it addresses the needs of those in the second camp.

A bit more of a ramble than intended. I actually set out to talk about the closest number to zero. It will have to wait for another day. Please add your thoughts below.

Calculus traditionally begins with a look at limits and continuity. In many courses this is the only time students will see discontinuous functions with removable discontinuities and most won’t see jump discontinuities elsewhere. So why do students have to learn about limits so early?

We tell them that they can’t do derivatives without limits. That, of course, is a lie. After a day or so of fussing with calculating derivatives using the limit definition we introduce them to shortcuts and algorithms. In fact, if someone came up to us and asked us out of the blue “what’s the derivative of [insert favorite elementary function here]”, we would not reach for the nearest limit.

We could resort to our favorite standby answer and say “because it’s going to be on the exam”. That doesn’t feel very satisfying though.

For me, limits are all about expectations. They are about predicting the way the world should work if everything is right. As x gets close to a certain value, does f(x) seem to hover at some level or does it fluctuate wildly like sin 1/x as x->0? Does it approach the same value from the left and the right or does there seem to be a jump as there is in f'(x) at 0 for f(x) = |x|? Somehow even |x| seems contrived to some students, but what about f(x)=x^(2/3). Near x=0 what happens to it’s derivative?

Limits let us know we aren’t in Kansas any more.

You certainly need limits if you’re going to prove any interesting theorems – but rather than expecting and entertaining questions like “when are we ever going to need this stuff”, let’s work to get our students to enjoy limits for what they are.

Calculus books are supposed to start with a review of PreCalculus. I think that’s a Lemma or something.

Here’s the problem with starting with PreCalculus – the people that need it don’t pay attention to it because they don’t know they need it yet. For instance – if you have this fraction:

6 x + 84
----------
2

You know you can’t just cancel the 6 and the 2. Well, maybe you don’t know it – but you certainly don’t have the patience in a Calculus class to write out:

2( 3 x + 42)
--------------- = 3 x + 42
2

You think you know what you’re doing and then you get to the quotient rule and you substitute in for the appropriate things in the formula

(f/g)' = f'g-g'f
-------
g^2

and you go ahead and end up canceling the g’s. Anyone who has graded calc homework or exams has seen that a bunch of times.

Most of what trips people up in Calculus is not the Calc but the PreCalc. But they aren’t ready to listen to the PreCalc the first week of Calculus because they think they know it already. After all, they’re in a Calculus class. They wouldn’t be in a calc class if they needed precalc.

So my preference is JIT PreCalc (Java programmers recognize that as “Just in Time”). When I’ve got your attention because you perceive a need, you will be ready for me to review the PreCalculus in place and not in this separate module that you’ll tune out.

Kimmy-the-wonderwife doesn’t expect much on Valentine’s Day. In fact, the first meeting of a new chapter of CocoaHeads (as in Cocoa tha Mac OS X framework) is tonight.

Over the years, my students have gotten to know Kim in many examples. She still remembers visiting one class and having a student ask her if she knew that I had thrown her into a pond in one example. She knew about that one. It was a related rates problem in which we looked at the ripples that were generated when she hit the water. She told the student that he shouldn’t worry, she was a very good swimmer. The student smiled and said “oh he told us that. That’s why he said he had to tie cement blocks to your ankles.”

I proposed to Kim on Valentines Day. Neither of us are very mushy about these things but she’s horrible at remembering dates so choosing this day makes it easy.

In any case – it is a very good Valentines Day. I’ve finished writing the first chapter of Head First Calculus and have fallen in love with the subject all over again. I miss being in front of a class and sharing the passion. But this is the next best thing. (Remind me to tell you some time about what I learned from Barry Manilow about teaching)

I’ve tried out the chapter on my nine year old and she seems to like it. The six year old likes the pictures.

Pictures? In a calculus book? Sure – we’re only 40 pages in and already have everything from ancient greeks to a man and a woman in bathing suits.

Writing a book requires sacrifice on the part of people around me. Today Kim gave me one of those special Valentine’s Day gifts that says she really cares in a way that flowers or chocolate couldn’t express (although chocolate would have been nice). She saw the look on my face as I printed out the first chapter and encouraged me to take the time to finish the book.

I have described myself alternately as a recovering Mathematician and a recovering Academic. Well, it seems that Mathematics was something I just couldn’t shake. I’ve fallen back off that wagon hard and am tilting at windmills in the ultimate display of folly: I’m writing a Calculus book.

Join me in my exploration of first year Calculus and other topics that may arise as I write “Head First Calculus” for O’Reilly Media